President Barack Obama’s surprise visit to Iraq on Tuesday came on the heels of a two-day visit to Turkey where he said: “The United States is not and never will be at war with Islam.”

Obama used his speech as a chance to continue his outreach to Muslims and to signal an approach to the region based more on pragmatism than ideology.

His words introduced what many here say is a sense of America’s new vision throughout the world. “A sense that, finally, an American leader understands,” wrote the New York Times.

According to the Washington Post-ABC News poll released this week, most Americans think Obama’s pledge to “seek a new way forward” with the Muslim world is an important goal, even as nearly half hold negative views about Islam and a sizable number say that even mainstream adherents to the religion encourage violence against non-Muslims,

An overwhelming 81 percent of Americans in the poll called it important for Obama to try to improve US relations with Muslim nations; 46 percent say it’s “very important.”

While slightly more than one in five expressed concerns that he’ll “go too far” in that effort, most by far, 65 percent, expect him to handle it about right.

In a CNN/Opinion Research Poll also released this week, 51 percent of Americans said the US should trust Muslim allies as it would other allies. Forty-eight percent said the US should trust Muslim allies less.

A large majority, 79 percent, said people in other countries will have a more positive view of the US because of Obama, while only 19 percent said it wouldw be more negative, according to the CNN poll.

Asked whether Obama has accomplished anything on his trip, 16 percent said that he had accomplished a great deal, while 45 percent thought he had accomplished a fair amount during his trip.

“Obama seems to have made a good first impression on the world stage, and Americans appear to have noticed,” said CNN Polling Director Keating Holland

But if the US has a leader who understands Islam, the poll said most Americans do not.